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I'm new to the turnstile package, and I'm trying to figure out how to get a negated double turnstile sign, like the one with the regular \nvDash, but that allows me to have arguments above and below the line, as with \sdtstile{\mathrm{a}}{b}.

Any ideas?

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Your best bet here, due to the size of the turnstiles, is to use the cancel package. Here is a minimal example:

enter image description here

\documentclass{article}
\usepackage{turnstile}% http://ctan.org/pkg/turnstile
\usepackage{cancel}% http://ctan.org/pkg/cancel
\begin{document}
$\sdtstile{\mathrm{a}}{b}$ \quad
$\cancel{\sdtstile{\mathrm{a}}{b}}$ \quad
$\bcancel{\sdtstile{\mathrm{a}}{b}}$
\end{document}

\cancel overprints a SW-NE line, while \bcancel (backwards/backslash) overprints a NW-SE line. As you can see, it prints the lines to match the content width. So, if you want the line to be a bit more diagonal, you could add some space around the turnstiles. However, I'm not sure what the exact contents is, and whether there are other objects around it, which would be affected by the spacing.

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